Translate to a system of equations and solve:

1) The sum of two numbers is 36. One number is 2 more than the other. Find the numbers.
Answer: (17,19)

2) Find two numbers whose sum is 66 and whose difference is 18.
Answer: (37,29)

These ones I am not sure of:

3) The difference between two numbers is 16. Three times the larger number is seven times the smaller. What are the numbers?

4) The difference between two numbers is 18. Twice the smaller number plus three times the larger is 74. What are the numbers?

3) Answer (12, 28)

4) Answer (4, 22)

What are the equations for numbers 3 and 4?

#3: Let the larger be x, the smaller be y

<The difference between two numbers is 18> ---> x - y = 16
<Three times the larger number is seven times the smaller> ---> 3x = 7y or 3x-7y = 0

#4 (same way)
x - y = 18
3x + 2y = 74

um of two numbers is

43
and the difference is
5
.

The average of two numbers is equal to twice the positive difference between the two numbers. If the larger number is 35, what is the smaller number.

To solve these types of problems, we need to set up a system of equations based on the given information and then solve the system to find the values of the unknown numbers. Let's take a look at each problem and break it down step by step.

3) The difference between two numbers is 16. Three times the larger number is seven times the smaller. What are the numbers?

Let's assume that the larger number is x and the smaller number is y.

From the given information, we can set up two equations:

Equation 1: x - y = 16 (since the difference between the numbers is 16)

Equation 2: 3x = 7y (since three times the larger number is seven times the smaller)

To solve this system, we can use the method of substitution or elimination.

Let's use substitution:

Rearrange Equation 2 to solve for x: x = (7y)/3

Substitute this value of x into Equation 1:

(7y)/3 - y = 16

Multiply both sides by 3 to get rid of the fractions:

7y - 3y = 48

Combine like terms:

4y = 48

Divide both sides by 4:

y = 12

Now substitute the value of y back into Equation 1:

x - 12 = 16

Add 12 to both sides:

x = 28

So the numbers are x = 28 and y = 12.

4) The difference between two numbers is 18. Twice the smaller number plus three times the larger is 74. What are the numbers?

Let's set x as the larger number and y as the smaller number.

From the given information, we can set up two equations:

Equation 1: x - y = 18 (since the difference between the numbers is 18)

Equation 2: 2y + 3x = 74 (since twice the smaller number plus three times the larger is 74)

Again, we can use the method of substitution or elimination.

Let's use elimination:

Multiply Equation 1 by 2 to make the coefficients of x the same:

2x - 2y = 36

Now, subtract Equation 1 from Equation 2:

(2y + 3x) - (2x - 2y) = 74 - 36

Simplify:

4y + 5x = 38

Now we have a new equation:

4y + 5x = 38

Subtract 5 times Equation 1 from this new equation:

(4y + 5x) - 5(x - y) = 38 - 90

Simplify:

4y + 5x - 5x + 5y = - 52

Combine like terms:

9y = - 52

Divide both sides by 9:

y = - 52 / 9

Now substitute the value of y into Equation 1:

x - (- 52 / 9) = 18

Multiply both sides by 9 to get rid of the fraction:

9x + 52 = 162

Subtract 52 from both sides:

9x = 110

Divide both sides by 9:

x = 110 / 9

So the numbers are x = 110/9 and y = - 52 / 9.